Standard-Deviation Normalization Proof
Fundamental
Role of Standard-Deviation Normalization in a Scale-Invariant
Statistical Framework
Author: Rowan Brad Quni-Gudzinas
Affiliation: QNFO Contact:
rowan.quni@qnfo.org ORCID: 0009-0002-4317-5604
ISNI: 0000 0005 2645 6062 DOI:
10.5281/zenodo.17216378 Publication Date: 2025-09-28
Version: 1.0
Given
Let \((\Omega, \mathcal{F}, P)\)be a
probability space, and let \(X\)be a
real-valued random variable defined on this space, representing a
physical observable. Assume \(X\)has a
well-defined mean \(\mu =
\mathbb{E}[X]\)and finite, positive variance \(\sigma^2 = \mathbb{E}[(X - \mu)^2] >
0\).
Consider the standardized random variable:
\[
Z := \frac{X - \mu}{\sigma}.
\]
Let \(\mathcal{M}\)be a statistical
manifold representing a family of probability distributions \(\{p(x|\theta)\}_{\theta \in \Theta}\),
where \(\theta\)are parameters. Assume
the family includes location-scale distributions of the form \(p(x|\mu, \sigma) = \sigma^{-1}
f((x-\mu)/\sigma)\), where \(f\)is a fixed probability density.
Goal
Demonstrate that standard-deviation normalization
(i.e., measuring observables in units of their inherent variability
\(\sigma\)) is fundamental to
constructing a **statistically unified, scale-invariant
description of reality**, as required by the Scale-Invariant
Epistemic Framework.
Derivation
Step
1: Standard-Deviation Normalization Defines Scale-Invariant
Coordinates
Define the dimensionless parameters:
\[
\eta := \frac{\mu}{\sigma}, \quad \xi := \log \sigma. \tag{1}
\]
These are the natural coordinates for the location-scale family. The
original parameters are recovered via \(\mu =
e^\xi \eta\), \(\sigma =
e^\xi\).
The transformation \((\mu, \sigma) \mapsto
(\eta, \xi)\)is precisely the mathematical operation of
standard-deviation normalization. The location
parameter \(\mu\)is expressed in units
of the scale parameter \(\sigma\), and
the scale parameter itself is expressed logarithmically.
Step
2: The Standardized Variable Z is Invariant under Scaling
Consider a global scale transformation of the observable \(X\):
\[
X \mapsto X' = \lambda X, \quad \lambda > 0.
\]
The mean and standard deviation transform as:
\[
\mu' = \lambda \mu, \quad \sigma' = \lambda \sigma.
\]
The standardized variable transforms as:
\[
Z' = \frac{X' - \mu'}{\sigma'} = \frac{\lambda X -
\lambda \mu}{\lambda \sigma} = \frac{X - \mu}{\sigma} = Z.
\]
Thus, \(Z\)is
invariant under global scaling of the original variable
\(X\).
Step
3: The Fisher Metric is Scale-Invariant in Normalized
Coordinates
For the location-scale family \(p(x|\mu,\sigma) = \sigma^{-1}
f((x-\mu)/\sigma)\), the standard Fisher metric components
are:
\[
g{\mu\mu} = \frac{I1}{\sigma^2}, \quad g_{\sigma\sigma} =
\frac{I2}{\sigma^2}, \quad g{\mu\sigma} = \frac{I_3}{\sigma^2},
\]
where \(I1, I2, I_3\)are constants
depending only on the base density \(f\).
Transforming to the normalized coordinates \((\eta, \xi)\)using the Jacobian:
\[
\frac{\partial(\mu, \sigma)}{\partial(\eta, \xi)} = \begin{pmatrix}
e^\xi & 0 \\ e^\xi \eta & e^\xi \end{pmatrix},
\]
the metric components become:
\[
\tilde{g}{\eta\eta} = g{\mu\mu} (e^\xi)^2 + 2 g_{\mu\sigma}
(e^\xi)(e^\xi \eta) + g_{\sigma\sigma} (e^\xi \eta)^2 =
\frac{1}{\sigma^2}(I1 + 2I3\eta + I2\eta^2) \cdot \sigma^2 = I1 +
2I3\eta + I2\eta^2, \tag{2a}
\]
\[
\tilde{g}{\xi\xi} = g{\sigma\sigma} (e^\xi)^2 = \frac{I_2}{\sigma^2}
\cdot \sigma^2 = I_2, \tag{2b}
\]
\[
\tilde{g}{\eta\xi} = g{\mu\sigma} (e^\xi)(e^\xi) + g_{\sigma\sigma}
(e^\xi \eta)(e^\xi) = \frac{1}{\sigma^2}(I3 + I2\eta) \cdot \sigma^2 =
I3 + I2\eta. \tag{2c}
\]
All components \(\tilde{g}_{ij}(\eta,
\xi)\)are independent of the scale parameter \(\sigma\) (or \(\xi\)). The metric is therefore
scale-invariant in the normalized coordinates \((\eta, \xi)\).
The line element is:
\[
ds^2 = \tilde{g}{ij}(\eta, \xi) \, d\theta^i d\theta^j = (I1 +
2I3\eta + I2\eta^2) d\eta^2 + 2(I3 + I2\eta) d\eta d\xi + I_2
d\xi^2. \tag{3}
\]
This metric is manifestly invariant under the transformation \(\eta \mapsto \lambda \eta\), \(\xi \mapsto \xi + \log \lambda\), which
corresponds to the original scaling \(X
\mapsto \lambda X\).
Step
4: Normalized Statistical Distances Are Scale-Invariant
The geodesic distance between two points \((\eta1, \xi1)\)and \((\eta2, \xi2)\)on the statistical
manifold is:
\[
d[(\eta1, \xi1), (\eta2, \xi2)] = \int_{\gamma} \sqrt{ds^2},
\]
where \(\gamma\)is the geodesic
path. Since the integrand \(ds^2\)is
invariant under the scaling transformation, the distance \(d\)is also invariant.
This implies that the statistical distinguishability
between two distributions, measured by their geodesic distance in the
normalized parameter space, is independent of the absolute scale of the
observables. It depends only on the dimensionless ratios \(\eta\)and the relative scale \(\Delta \xi = \xi2 - \xi1\).
Step
5: Connection to Physical Quantities and Information
Theory
The standardized variable \(Z = (X -
\mu)/\sigma\)represents the **number of standard
deviations** \(X\)deviates from
its mean. This is a **dimensionless measure of statistical
significance**.
The Signal-to-Noise Ratio (SNR) is a fundamental
quantity in information theory and physics:
\[
\text{SNR} = \frac{\mu}{\sigma} = \eta.
\]
This is precisely the normalized location parameter \(\eta\).
The Coefficient of Variation (CV) is another
scale-invariant measure:
\[
\text{CV} = \frac{\sigma}{\mu} = \frac{1}{\eta}.
\]
The Fisher information matrix, when expressed in
normalized coordinates, yields dimensionless components \(\tilde{g}_{ij}\), representing the
information content per unit of standard deviation.
Step 6:
Universality and Critical Phenomena
In the context of critical phenomena, the correlation function \(\langle \mathcal{O}(0) \mathcal{O}(r)
\rangle\)is often normalized by the variance \(\langle \mathcal{O}^2 \rangle\):
\[
G(r) = \frac{\langle \mathcal{O}(0) \mathcal{O}(r) \rangle}{\langle
\mathcal{O}^2 \rangle}.
\]
This normalized function \(G(r)\)exhibits universal scaling behavior
\(G(r) \sim r^{-\eta}\)near critical
points, where \(\eta\)is a universal
critical exponent (dimensionless). The normalization by variance ensures
that \(G(r)\)is dimensionless and
scale-invariant.
Step
7: Implications for the Scale-Invariant Epistemic
Framework
The scale invariance of the Fisher metric in normalized coordinates
\((\eta, \xi)\)demonstrates that:
**Standard-deviation normalization provides the correct
coordinate system** for describing statistical manifolds in a
scale-invariant manner.
**Statistical distances and distinguishability measures become
scale-invariant** when computed using normalized parameters.
**Physical laws expressed in terms of normalized
quantities** (like SNR, CV, correlation functions normalized by
variance) automatically satisfy the principle of universal scale
invariance.
The standard deviation \(\sigma\)acts as the natural unit
for measuring the variability of an observable \(X\), consistent with the epistemic
principle that knowledge is limited by inherent fluctuations.
Conclusion
**Theorem (Fundamental Role of Standard-Deviation
Normalization).** Standard-deviation normalization, achieved by
expressing observables as \(Z = (X -
\mu)/\sigma\)and parameters in dimensionless form like \(\eta = \mu/\sigma\), is mathematically
necessary for constructing a **scale-invariant statistical
description** of reality. This normalization renders the Fisher
information metric and associated geometric quantities (distances,
curvatures) invariant under global scale transformations of the
underlying observables. Consequently, all physically meaningful
statistical measures (e.g., SNR, CV, normalized correlation functions)
become dimensionless and scale-invariant, providing the unified,
observer-independent framework required by the Scale-Invariant Epistemic
Framework.
This formalizes the principle that reality’s statistical structure is
fundamentally captured by **dimensionless ratios relative to
intrinsic variability**, with the standard deviation \(\sigma\)serving as the fundamental “ruler”
for measuring fluctuations.
Q.E.D.